Problem: Alex earns a $\$35{,}000$ salary in the first year of his career. Each year, he gets a $3\%$ raise. Which expression gives the total amount Alex has earned in his first $n$ years of his career? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{35{,}000(1-1.03^n)}{-0.03}$ (Choice B) B $\dfrac{35{,}000(1-0.97^n)}{0.03}$ (Choice C) C $\dfrac{1.03(1-35{,}000^n)}{35{,}000}$ (Choice D) D $\dfrac{0.97(1-35{,}000^n)}{35{,}000}$
Answer: Notice that Alex's salaries over the years form a geometric sequence. The total amount Alex earns after $ n$ years is the ${\text{sum}}$ of the first $n$ terms in the sequence. This is called a geometric series. This is the formula for that sum: $ S={a}\left(\dfrac{1-{r}^{ n}}{1-{r}}\right)$ where ${a}$ is the first term and ${r}$ is the common ratio. We can use this formula, along with the given information, to find the expression for the sum, $ S$. Using the given information We are given that Alex earns a ${\$35{,}000}$ salary in the first year of his career. This is the first term $ a$. We are given that he gets a ${3\% \text{ raise}}$ each year. So we'll use a common ratio of ${1.03}$ for $ r$. We are interested in the first ${n}$ years, so the number of terms is $ {n}$. We want an expression for the total amount he earns. This is the sum $ S$. Writing the sum $ S={35000} \cdot \dfrac{1-\left({1.03}\right)^{{n}}}{1-\left({1.03}\right)}$ Answer The total amount Alex has earned in his first $n$ years of his career is: $\dfrac{35{,}000(1-1.03^n)}{-0.03}$